The proposed prior in (REF ) is employed to model the sparsity pattern of \(\mathbf {G}\) .
Different from the previous works [1]}, [2]} that only adopted Gaussian-gamma pairs for modeling the column-wise sparsity, the prior in (REF ) utilizes the product rule of probability to further encode the element-wise sparsity information. The proposed prior has a clear physical interpretation. Particularly, \(\xi (n,k)^{-1}\)  can be interpreted as the power of each element \(\mathbf {G}(n,k)\) ,
while \(\eta _k^{-1}\)  has a physical interpretation as the power of each column in \(\mathbf {G}\) .
Therefore, if \(\eta _k^{-1}\)  is learnt to approach zero, regardless of \(\xi (n,k)\) , the corresponding columns in the factor matrices play no role in channel modeling and thus can be pruned out by thresholding. Similarly, for a nonzero column (\(\eta _{k}^{-1}\ne 0\) ), \(\mathbf {G}(n,k)\)  would be shrunk to zero as \(\xi (n,k)^{-1}\)  goes to zero. Therefore, the proposed prior in (REF ) can simultaneously promote element-wise sparsity of the channel and low-rank property of tensor data, which mimic the sparsity constraint and the last regularization term in (REF ).
Note that the conventional way to promote both element-wise and column-wise sparsity is to use hierarchical Bayesian [3]}. However, the resulting penalty is less capable in enforcing a structural low-rank to the tensor. In this context, multiplying two priors serves as a viable approach since both hyperparameters influence \(\mathbf {G}\)  directly.
